$${{{\mathbb Z}_2}{{\mathbb Z}_4}}$$ -linear codes: generator matrices and duality

Abstract

A code C is Z 2 Z 4 -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y ) is a binary linear code (respectively, a quaternary linear code). In this paper Z 2 Z 4 -additive codes are studied. Their corresponding binary images, via the Gray map, are Z 2 Z 4 -linear codes, which seem to be a very distinguished class of binary group codes. As for binary and quaternary linear codes, for these codes the fundamental parameters are found and standard forms for generator and parity-check matrices are given. In order to do this, the appropriate concept of duality for Z 2 Z 4 -additive codes is defined and the parameters of their dual codes are computed.

Partial permutation decoding for binary linear Hadamard codes

“…Note 1 Proposition 4.1 is also true if we define the matrix M (v) taking as vector v any of the last m rows of M instead of the last one as in (2). Note 2 The bound f H m+1 for H m+1 cannot be achieve recursively from an s-PD-set for H m .…”

“…An alternative permutation decoding algorithm for Z 2 Z 4 -linear codes [2] is described in [1]. In particular, it can be applied to Hadamard Z 2 Z 4 -linear codes.…”

The Structure of Z2Z2s-Additive Codes: Bounds on the Minimum Distance

“…Considering all these parameters, C is called a Z 2 Z 4 −additive code of type (α, β ; γ, δ ; κ). We say that the binary image C = φ 1 (C ) is a Z 2 Z 4 −linear code of length n = α + 2β [2].…”

“…Most of the concepts on Z 2 Z 4 −additive codes have been described in [2]. In this paper, we study Z 2 Z 2 s −additive codes for s > 1, where Z 2 Z 4 −additive codes are a special case.…”

Efficient representation of binary nonlinear codes: constructions and minimum distance computation

“…On the other hand, if C is nonlinear, then a solution would be to know whether it has another structure or not. For example, there are binary codes which have a Z 4 -linear or Z 2 Z 4 -linear structure and, therefore, they can also be compactly represented using a quaternary generator matrix [4,12]. In general, binary codes without considering any such structure can be seen as a union of cosets of a binary linear subcode of C [1].…”